Complex analysis in number theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press
1995
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 187 S. |
| ISBN: | 0849328667 |
Staff View
MARC
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| 100 | 1 | |a Karacuba, Anatolij A. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Complex analysis in number theory |c Anatoly A. Karatsuba |
| 264 | 1 | |a Boca Raton [u.a.] |b CRC Press |c 1995 | |
| 300 | |a IX, 187 S. | ||
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Record in the Search Index
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| adam_text | _ V
Contents
Introduction 1
Chapter 1. The Complex Integration Method and Its
Application in Number Theory 8
1. Generating Functions in Number Theory 8
1.1 Dirichlet s series 8
1.2 Sum functions 11
2. Summation Formula 13
2.1 Perron s formula 13
2.2 Expressing Chebyshev s function in terms of the
integral of the logarithmic derivative of Riemann s
zeta function 14
3. Riemann s Zeta Function and Its Simplest Properties . . 15
3.1 The functional equation 15
3.2 Riemann s hypotheses 17
3.3 The simplest theorems on the zeros of ((s) 18
3.4 Expressing Chebyshev s function as a sum over the
complex zeros of £(s) 19
3.5 The asymptotic law of distribution of prime numbers 20
3.6 Riemann s hypothesis concerning the complex
zeros of ((s) and the problem of the theory of prime
numbers 21
3.7 Theorem on the uniqueness of £(s) 23
3.8 Proofs of the simplest theorems on the complex
zeros of ((s) 24
vi Contents
Chapter 2. The Theory of Riemann s Zeta Function 31
1. Zeros on the Critical Line 31
1.1 Hardy s theorem 31
1.2 Theorems of Hardy and Littlewood 31
1.3 Hardy s function and Hardy s method 32
1.4 Titchmarsh s discrete method 35
1.5 Selberg s theorem 35
1.6 Estimates of Selberg s constant 36
1.7 Moser s theorems 37
1.8 Selberg s hypothesis 38
1.9 Zeros of the derivatives of Hardy s function .... 39
1.10 The latest results 40
1.11 Distribution of zeros in the mean 41
1.12 Density of zeros on the critical line 41
1.13 The zeros of £(s) in the neighborhood of the critical
line 42
2. The Boundary of Zeros 43
2.1 De la Vallee Poussin theorem 43
2.2 Littlewood s theorem 43
2.3 The relationship between the boundary of zeros
and the order of growth of ((s) in the neighbor¬
hood of unit line 44
2.4 Vinogradov s method in the theory of ((s) and Chu
dakov s theorems 45
2.5 Vinogradov s theorem 46
3. Approximate Equations of the £(s) Function 47
3.1 Partial summation and Euler s summation formula 47
3.2 The simplest approximation of £(s) 49
3.3 The approximation of a trigonometric sum by a
sum of trigonometric integrals 50
3.4 Asymptotic calculations of a certain class of trigono¬
metric integrals 57
3.5 Approximation of a trigonometric sum by a more
concise sum 66
3.6 Approximate equations of the £(s) function .... 69
3.7 On trigonometric integrals 73
Contents vii
4. The Method of Trigonometric Sums in the Theory of the
C(s) Function 77
4.1 The mean value of the degree of the modulus of a
trigonometric sum 77
4.2 Simple lemmas 78
4.3 The basic recurrent inequality 83
4.4 Vinogradov s mean value theorem 89
4.5 The estimate of the zeta sum and its consequences 91
4.6 The current boundary of zeros of £(s) and its corol¬
laries 98
5. Density Theorems 100
5.1 Bertrand s postulate and Chebyshev s theorem . . 100
5.2 Hoheisel s method 100
5.3 Density of zeros of £(s) 102
5.4 Density theorems 103
5.5 Proof of Huxley s density theorem 104
5.6 Three problems of the number theory solvable by
Hoheisel s method 120
6. The Order of Growth of C(s) in a Critical Strip 122
6.1 The problem of Dirichlet s divisors 123
6.2 Lindelof s hypothesis 124
6.3 Equivalents of Lindelof s hypothesis 125
6.4 The order of growth of |C(| + t )I 126
6.5 Vinogradov s method in Dirichlet s multi dimen¬
sional divisor problem 127
6.6 Omega theorems 130
7. Universal Properties of the ((s) Function 130
7.1 Bohr s theorems 130
7.2 Voronin s theorems 132
7.3 Theorem on the universal character of ((s) .... 134
7.4 More on the universal character of C(s) 135
8. Riemann s Hypothesis, Its Equivalents, Computations . . 135
8.1 Mertens hypothesis 136
8.2 Turan s hypothesis and its refutation 137
8.3 A billion and a half complex zeros of C(s) 138
8.4 Computations connected with ((s) 138
viii Contents
8.5 Functions resembling £(s) but having complex
zeros on the right of the critical line 139
8.6 Epstein s zeta functions 140
8.7 A new approach to the problem of zeros, lying on
the critical line, of some Dirichlet series 141
Chapter 3. Dirichlet L Functions 147
1. Dirichlet s Characters 147
1.1 Definition of characters 147
1.2 Principal properties of characters 148
2. Dirichlet X Functions and Prime Numbers in Arithmetic
Progressions 149
2.1 Definition of X functions 149
2.2 The functions ir(x;k,l) and ij (x;k,l) 150
2.3 Dirichlet s theorem on primes 150
3. Zeros of Z Functions 152
3.1 The boundary of zeros. Page s theorems 152
3.2 Siegel s theorem 153
3.3 Zeros on the critical line 153
4. Real Zeros of Z Functions and the Number of Classes of
Binary Quadratic Forms 154
4.1 Binary quadratic forms and the number of classes . 154
4.2 Dirichlet s formulas 156
4.3 Gauss problem and Siegel s theorem 156
4.4 Prime numbers in arithmetic progressions 157
5. Density Theorems 158
5.1 Linnik s density theorems 158
5.2 Density theorems of a large sieve and the Bombieri
Vinogradov theorem 158
5.3 Current density theorems 160
5.4 Proof of Vinogradov s theorem on three prime num¬
bers based on the ideas of Hardy Littlewood Linnik 160
6. Z Functions and Nonresidues 163
6.1 The concept of a nonresidue 163
6.2 Vinogradov s hypothesis 163
Contents ix
6.3 Lindelof s generalized hypothesis and a nonresidue 164
6.4 The zeros of the Z functions and nonresidues . . . 164
7. Approximate Equations 165
7.1 Stating the problem 165
7.2 Lavrik s general theorem 165
8. On Primitive Roots 168
8.1 The concept of a primitive root 168
8.2 Artin s hypothesis 168
8.3 Hooley s conditional theorem 168
References 170
Author Index 183
Subject Index 185
|
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| id | DE-604.BV010233169 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:48:57Z |
| institution | BVB |
| isbn | 0849328667 |
| language | English |
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| owner_facet | DE-12 DE-91 DE-BY-TUM DE-20 DE-11 DE-188 |
| physical | IX, 187 S. |
| publishDate | 1995 |
| publishDateSearch | 1995 |
| publishDateSort | 1995 |
| publisher | CRC Press |
| record_format | marc |
| spelling | Karacuba, Anatolij A. Verfasser aut Complex analysis in number theory Anatoly A. Karatsuba Boca Raton [u.a.] CRC Press 1995 IX, 187 S. txt rdacontent n rdamedia nc rdacarrier Funktionentheorie - Zahlentheorie Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Analytische Zahlentheorie (DE-588)4001870-2 gnd rswk-swf Analytische Zahlentheorie (DE-588)4001870-2 s DE-604 Zahlentheorie (DE-588)4067277-3 s Funktionentheorie (DE-588)4018935-1 s DE-188 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006802624&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Karacuba, Anatolij A. Complex analysis in number theory Funktionentheorie - Zahlentheorie Zahlentheorie (DE-588)4067277-3 gnd Funktionentheorie (DE-588)4018935-1 gnd Analytische Zahlentheorie (DE-588)4001870-2 gnd |
| subject_GND | (DE-588)4067277-3 (DE-588)4018935-1 (DE-588)4001870-2 |
| title | Complex analysis in number theory |
| title_auth | Complex analysis in number theory |
| title_exact_search | Complex analysis in number theory |
| title_full | Complex analysis in number theory Anatoly A. Karatsuba |
| title_fullStr | Complex analysis in number theory Anatoly A. Karatsuba |
| title_full_unstemmed | Complex analysis in number theory Anatoly A. Karatsuba |
| title_short | Complex analysis in number theory |
| title_sort | complex analysis in number theory |
| topic | Funktionentheorie - Zahlentheorie Zahlentheorie (DE-588)4067277-3 gnd Funktionentheorie (DE-588)4018935-1 gnd Analytische Zahlentheorie (DE-588)4001870-2 gnd |
| topic_facet | Funktionentheorie - Zahlentheorie Zahlentheorie Funktionentheorie Analytische Zahlentheorie |
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